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Collinear Points:
Collinear points are points that lie on the same straight line. In other words, if three or more points lie on the same line, they are said to be collinear.
Given Points:
The given points are (t, 2t), (-2, 6), and (3, 1).
Checking for Collinearity:
To check if the given points are collinear, we can use the slope formula or the distance formula.
Slope Formula:
The slope between two points (x1, y1) and (x2, y2) is given by the formula:
m = (y2 - y1) / (x2 - x1)
Using this formula, we can find the slopes of the line formed by the points (t, 2t) and (-2, 6), and the line formed by the points (-2, 6) and (3, 1).
Line 1:
Slope of the line between (t, 2t) and (-2, 6):
m1 = (6 - 2t) / (-2 - t)
Line 2:
Slope of the line between (-2, 6) and (3, 1):
m2 = (1 - 6) / (3 - (-2))
If the slopes of both lines are equal, then the points are collinear. Let's equate the slopes and solve for t.
Equate the slopes:
(6 - 2t) / (-2 - t) = (1 - 6) / (3 - (-2))
Simplifying the equation:
(6 - 2t) / (-2 - t) = -5 / 5
(6 - 2t) / (-2 - t) = -1
Cross multiplying:
(6 - 2t) * 5 = (-2 - t) * (-1)
30 - 10t = 2 + t
Combining like terms:
-10t - t = 2 - 30
-11t = -28
Solving for t:
t = (-28) / (-11)
t = 28 / 11
t = 8/11
Conclusion:
Therefore, if the points (t, 2t), (-2, 6), and (3, 1) are collinear, then t = 8/11.
Collinear points are points that lie on the same straight line. In other words, if three or more points lie on the same line, they are said to be collinear.
Given Points:
The given points are (t, 2t), (-2, 6), and (3, 1).
Checking for Collinearity:
To check if the given points are collinear, we can use the slope formula or the distance formula.
Slope Formula:
The slope between two points (x1, y1) and (x2, y2) is given by the formula:
m = (y2 - y1) / (x2 - x1)
Using this formula, we can find the slopes of the line formed by the points (t, 2t) and (-2, 6), and the line formed by the points (-2, 6) and (3, 1).
Line 1:
Slope of the line between (t, 2t) and (-2, 6):
m1 = (6 - 2t) / (-2 - t)
Line 2:
Slope of the line between (-2, 6) and (3, 1):
m2 = (1 - 6) / (3 - (-2))
If the slopes of both lines are equal, then the points are collinear. Let's equate the slopes and solve for t.
Equate the slopes:
(6 - 2t) / (-2 - t) = (1 - 6) / (3 - (-2))
Simplifying the equation:
(6 - 2t) / (-2 - t) = -5 / 5
(6 - 2t) / (-2 - t) = -1
Cross multiplying:
(6 - 2t) * 5 = (-2 - t) * (-1)
30 - 10t = 2 + t
Combining like terms:
-10t - t = 2 - 30
-11t = -28
Solving for t:
t = (-28) / (-11)
t = 28 / 11
t = 8/11
Conclusion:
Therefore, if the points (t, 2t), (-2, 6), and (3, 1) are collinear, then t = 8/11.
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